Appeal to Probability Fallacy: When It Could Happen Becomes It Will

The fallacy is a slide between three words that ordinary speech keeps loose: possible, probable and certain. A possible event is one that cannot be ruled out. A probable event is one with odds high enough to act on. A certain event is one that will occur. The appeal to probability takes the first and treats it as the third, usually without the middle step ever being stated. Nothing in "it could happen" tells you how often it does, and a conclusion that needs the frequency cannot be drawn from the bare possibility.

The slide feels natural because the mind does not weigh small probabilities in proportion. Kahneman and Tversky showed in 1979 that most people prefer a one-in-a-thousand chance of 5,000 to a sure 5, though the two have the same expected value, and that when every option is a long shot people pick the one with the biggest prize rather than the one with the best odds: 73 percent of their respondents chose a 0.1 percent chance of 6,000 over a 0.2 percent chance of 3,000. A possibility, once imagined, takes up more room than its odds deserve. Rottenstreich and Hsee found in 2001 that this flattening is strongest for outcomes with a strong emotional charge: for a painful electric shock, the price people would pay to avoid a 1 percent chance and a 99 percent chance was almost the same. Emotion makes the possibility vivid, and vividness does the rest.

The fallacy runs in both directions. In the hopeful direction it builds plans on a win that is possible and nothing more: the lottery, the viral post, the discovery that will make the thesis. In the fearful direction it turns a conceivable harm into an expected one: the plane that could crash, the stranger who could be dangerous, the symptom that could be serious. Both versions skip the same step. A possible good and a possible harm each need an estimate of how likely they are before they can tell you what to do, and the estimate is the part that the fallacy leaves out.

The line between the fallacy and sound reasoning about risk is the number. "This could happen, so prepare for it" is often sensible when the preparation is cheap and the probability is real; that is what Murphy's law means in engineering, where it began as a rule for designing against failure. "This could happen, so it will" is the fallacy. The two are easy to confuse because they begin with the same premise, and they can only be told apart by asking what the probability actually is and what the conclusion is actually claiming.

The appeal to probability is a label from informal logic, not a finding from a laboratory, and no single experiment established it. The habit of cataloging bad arguments goes back to Aristotle's Sophistical Refutations, which listed thirteen ways an argument can seem to prove what it does not, and modern catalogs such as the Stanford Encyclopedia of Philosophy's entry on fallacies and Gary Curtis's Fallacy Files carry the tradition on. The appeal to probability does not appear in Aristotle's list; it is a modern name for a pattern that was always there, the confusion of possibility with certainty.

What psychology supplies is the reason the pattern is so easy to fall into. In the 1979 paper that introduced prospect theory, Daniel Kahneman and Amos Tversky gave students and university faculty in Israel a series of choices between gambles with outcomes in Israeli pounds. Problem 14 offered a one-in-a-thousand chance of 5,000 against 5 for sure: 72 percent of the 72 respondents took the chance, choosing what the authors called a lottery ticket over its own expected value. The mirror problem offered a one-in-a-thousand chance of losing 5,000 against a sure loss of 5, and 83 percent paid the sure 5, as if buying insurance. In Problem 8, which set a 0.1 percent chance of 6,000 against a 0.2 percent chance of 3,000, 73 percent of 66 respondents chose the larger prize at half the odds. The authors' conclusion was that very low probabilities are generally overweighted, and that when winning is possible but not probable people choose by the size of the prize.

Yuval Rottenstreich and Christopher Hsee extended the finding in 2001 to outcomes with emotional weight. The median participant in their study would pay 7 dollars to avoid a 1 percent chance of a short electric shock and 10 dollars to avoid a 99 percent chance of it, while for a 20 dollar cash penalty the medians were 1 dollar and 18 dollars. For the shock, probability almost stopped mattering: the possibility was the thing being paid for. The appeal to probability fallacy is what this looks like when it is written down as an argument.

The pattern shows up wherever a plan or a fear rests on a possibility that has never been given odds. The rows are illustrations, not findings from the cited studies.

The fallacy leaves out one step, so the counter is to put it back.

• Attach a number. When an argument rests on something that could happen, ask how often it does. A rough estimate, even one in ten or one in ten thousand, turns a possibility into something that can be weighed.
• Separate prediction from precaution. "It will happen" and "it is worth preparing for" are different claims with different costs. Many sound decisions prepare for events that will probably not occur; the fallacy is claiming they will.
• Watch for the hidden promotion. Possible, likely, bound to and certain are often used interchangeably in a single conversation. Notice when a claim enters as possible and leaves as certain without anything happening in between.
• Run the argument in the other direction. If a possible win justifies a plan, a possible loss of the same odds justifies the opposite plan. An argument that works for any possibility proves nothing about any of them.
• Compare with the alternatives. A risk that cannot be ruled out is rarely the only one in play; canceling the flight puts you on the road. Put the possibilities side by side with their odds before acting on one of them.

The appeal to probability is a mistake about the step from possibility to certainty. Several neighbors are mistakes about probability itself.

• It is not Murphy's law, which as a planning rule says to design for the failures that can happen, and as a feeling says the world is arranged against you. The fallacy is Murphy's law read as a forecast: because the failure can happen, it will.
• It is not neglect of probability, in which the odds of an outcome barely register because the outcome is emotionally charged. Neglect of probability is the state of mind; the appeal to probability is the argument that results when the missing odds are replaced with certainty.
• It is not the gambler's fallacy, the expectation that a run of one outcome makes the other due. The gambler has a probability in mind and gets it wrong; the appeal to probability never consults one.